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A Hierarchical Geometric Framework to Design Locomotive Gaits for Highly Articulated Robots

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A Hierarchical Geometric Framework to Design Locomotive Gaits for Highly Articulated Robots Robotics: Science and Systems 2019 Freiburg im Breisgau, June 22-26, 2019 A Hierarchical Geometric Framework to Design Locomotive Gaits for Highly Articulated Robots Baxi Chong∗, Yasemin Ozkan Aydin∗, Guillaume Sartorettiy, Jennifer M Rieser∗, Chaohui Gongy, Haosen Xingy, Howie Choset y and Daniel I Goldman ∗ ∗Georgia Institute of Technology, yCarnegie Mellon Institution Abstract—Motion planning for mobile robots with many degrees-of-freedom (DoF) is challenging due to their high- dimensional configuration spaces. To manage this curse of di- mensionality, this paper proposes a new hierarchical framework that decomposes the system into sub-systems (based on shared capabilities of DoFs), for which we can design and coordinate motions. Instead of constructing a high-dimensional configuration space, we establish a hierarchy of two-dimensional spaces on which we can visually design gaits using geometric mechanics tools. We then coordinate motions among the two-dimensional spaces in a pairwise fashion to obtain desired robot locomotion. Further geometric analysis of the two-dimensional spaces allows us to visualize the contribution of each sub-system to the locomotion, as well as the contribution of the coordination among the sub-systems. We demonstrate our approach by designing gaits for quadrupedal robots with different morphologies, and Fig. 1. Three link swimmer with four cubic legs CAD diagram of the experimentally validate our findings on a robot with a long robot (a) top and (b) side view. (c) Physical model. The upper back and lower back are actuated by two AX12 servos (labeled as 1 and 2) and the legs are actuated back and intermediate-sized legs. actuated by XL320 servos. The legs have rectangular shape (24x24x45 mm) and the body is always in contact with ground via blocks (55x12x45 mm) I. INTRODUCTION attached to the body segments. High degree-of-freedom (HDoF) mobile robots offer ad- vantages over conventional wheeled vehicles in difficult-to- convergence is attained. Once this process has converged, negotiate terrains [3, 17, 21, 25]. However, the benefit of many we further show how GM tools allow us to visualize the degrees of freedom also poses a challenge: how to coordinate contribution of each sub-system to locomotion, as well as all of the DoFs to produce purposeful motion. One approach that of the coordination of every pair of sub-systems. In this to address this so-called curse of dimensionality is to couple way, the proposed hierarchical framework provides a motion groups of the DoFs into one or two DOFs thereby producing a planning tool to visually and intuitively design gaits for HDoF lower dimensional configuration space [5, 10, 15, 16, 24, 2, 6]. systems. We term these lower dimensional systems as sub-systems. Sub- We apply our framework to study the motion of systems can be formed based on observations of biological a quadrupedal robot with a long articulated body and systems, empirical experiments, or intuition. The approach intermediate-sized legs (Fig. 1), a morphology incorporating described in this paper first plans for motions within every features from both lizards and snakes. Here, we show how sub-system, and then for coordination among the sub-systems. our framework provides insight into the role of leg contacts Specifically, we propose a hierarchical framework to decom- and body undulation in quadruped locomotion. Previous work pose the motion planning problem in the full configuration has shown that the undulation of an elongated body improves space of a robot, into i) a series of simpler motion planning the stride length of animals and robots [11, 27, 8]. A more problems in the low-dimensional spaces associated with the comprehensive review on quadrupedal gait design can be different sub-systems, and ii) the coordination of these sub- found in [27]. There are two common modes of body un- systems. dulation: standing waves (lizard-like) [11, 27], and traveling Our framework relies on a motion planning approach, waves propagated from head to tail (snake-like) [8]. In the first namely geometric mechanics (GM), which offers powerful mode, the body curvature is maximized to increase the self- tools to design, visualize, and analyze a wide range of lo- propulsion generated by leg-ground interaction. In the second comotor behaviors [4, 7, 14]. We rely on GM tools to visually mode, the body undulation can generate self-propulsion even design the motion of each subsystem with respect to our without legs. Interestingly, we find that the optimized whole- locomotive goals (here, to maximize stride displacement), and body motion (i.e., motion in the full configuration space) for to design the coordination of pairs of sub-systems. By rely- our robot is a hybrid combination of the lizard- and snake-like ing on a coordinate-descent-like approach, we then optimize modes, such that the self-propulsion from leg-ground contact locomotion by iterating over every pair of sub-systems until and that from body-ground contact are properly coordinated. 1 maps the changes in internal shape variables (joint angles) to changes in group variables (position and orientation) of the robot. The interaction between a robot and an environment is often difficult to model, which is typically the case when the system locomotes on a granular medium. In such a case, calculating the local connection matrix A can be quite challenging. Instead, we numerically derive A using resistive force theory (RFT) [13, 22, 26] to model the contact. Prior work [8] has shown that numerically derived local connections using Fig. 2. Example height functions and gaits in different shape spaces granular RFT can effectively predict movements in granular (a) The height function in a Euclidean shape space corresponds to motions of an 8-link snake robot slithering in the forward direction on the surface media. In this paper, we model the robot-ground contact by of 6mm plastic particles. The purple circle represents the optimal gait in the poppy seed RFT equations [27], from which we numerically corresponding shape space. (b) The height function in a torus shape space derive the local connections. corresponds to the forward motion of a robot with a long actuated body and intermediate-sized legs on the surface of ∼ 1 mm poppy seeds. The 2) Connection vector fields and height functions: Each row solid purple curve represents a sample gait in the corresponding shape space. of local connection matrix A corresponds to a component Orange lines represent the assistive lines to form closed loops with the gait path in the unfolded torus shape space. The area in the solid purple shadow direction of the body velocity and therefore gives rise to a represents the area where the gait path and the assistive lines form a clockwise connection vector field. The body velocities in the forward, loop; the area in the dashed purple shadow represents the area where the gait lateral and rotational directions are respectively computed as path and the assistive lines form a counterclockwise loop. Red, white and black indicate positive, zero and negative values in height function respectively. the dot product of connection vector fields and the shape velocity r_. A shape velocity r_ along the direction of the vector field yields the largest possible body velocity, while a shape We validate our theoretical predictions through a series of velocity r_ orthogonal to the field produces zero body velocity. experiments where a 10-DoF robot locomotes on granular A gait can be represented as a closed curve in the corre- material. sponding shape space. The displacement resulting from a gait, The paper is structured as follows: Section II introduces @χ, can be approximated by 0 1 geometric mechanics and presents our framework to design ∆x Z whole-body motion; Section III applies our methods to study B C the motion design of a robot with a long articulated body and B∆yC = A(r)dr: (2) @ A @χ four legs; Section IV discusses the significance and concludes ∆θ our work. According to Stokes’ Theorem, the line integral along a II. METHOD closed curve @χ is equal to the surface integral of the curl of A. Review of geometric mechanics A(r) over the area enclosed by @χ: Z ZZ In this section, we provide a concise overview of the A(r)dr = r × A(r)dr1dr2; (3) geometric tools needed for this paper. For a more detailed @χ χ and comprehensive review, we refer readers to [4, 7, 14]. where χ denotes the area enclosed by @χ. The curl of the The gait design tools of geometric mechanics separate the connection vector field, r × A(r), is referred to as the height configuration space of a system into two spaces: the position function (Fig. 2a). The three rows of the vector field A(r) can space and the shape space. The position space denotes the thus produce three height functions in the forward, lateral and location of a system relative to a world frame, while the shape rotational direction, respectively. space denotes the internal configuration (internal shape) of the system. Geometric mechanics seeks to establish a functional With the above derivation, we simplify the gait design prob- relationship between velocities in these spaces; this functional lem to drawing a closed path in Euclidean shape space. The relationship is often called a connection, and it shares many displacements can be approximated by the volume enclosed by properties with the Jacobian of a robotic manipulator. the gait path. For example, in Fig. 2a, we show an example 1) Kinematic reconstruction equation: Principally, in kine- height function in Euclidean shape space corresponding to matic systems where there is no drift in the system, the the motion in the forward direction of an 8-link snake robot equations of motion [14] reduce to slithering on the surface of 6mm plastic particles.
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